AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple adom sample: A sample whch evey subject the populato has the same chace to be selected. : The Radom Vaable deote the heght of a adult male we wll choose adomly fom the populato So ~ N(, ) : the dstbuto of a adomly selected subject s the populato dstbuto. Theoem Samplg fom the omal populato Let..d.,,..., ~ (, ) N, whee..d stads fo depedet ad detcally dstbuted. ~ N(, ). ( ) S ( ) ~ (Ch Squae dstbuto wth ( ) degees of feedom), *Remde: The Sample vaace S s defed as: S ( ) *Def : The Ch-squae dstbuto s a specal gamma dstbuto (*** Please fd out whch oe t s.)..d. *Def : Let Z, Z,..., Z ~ N (0,), The W Z ~ k k k
3. & S ae depedet. 4. Z ~N(0,) T S ~ t 5. (t dstbuto wth (-) degees of feedom) *Def. of t-dstbuto (Studet s t-dstbuto, fst toduced by Wllam Sealy Gosset ) Let Z ~ N (0,), depedet. The, T Z W k ~ t k W, whee Z& ~ k W ae
Wk: Wllam Sealy Gosset (Jue 3, 876 Octobe 6, 937) s best kow by hs pe ame Studet ad fo the Studet's t-dstbuto. *Poof of 5 We kow that W ( ) S Futhemoe, sce ~ Z ~ (0,) N ad & S ae depedet, thus Z& ae depedet. Theefoe by the defto of t-dstbuto, Z W ~ t W *We wll fst pove # & #3 fo the specal case of =. I that case, we have: Idepedet ~ N(, ) ~ N(, ) ( ) ( ) S ( ) *# ( ) S S ( ) W Z ~ N(0,) *You ca pove ths usg the pvotal quatty method W Z ~ (usg the d Defto of Ch Squae *#3 Dstbuto.) If we ca show ad ae depedet 3
the we have pove that ad S ae depedet. Appoach : p.d.f. f, (, ) f ( ) f ( ) Appoach: m.g.f. M ( t, t ) M ( t ) M ( t ), The jot m.g.f. of ad Y s defed as: M t t E e t ty Y, (, ) ( ) Note: We have doe ths aleady. Addtoal Questos ad Solutos Q. Pove Soluto ES ( ) fo ay dstbuto/populato. S ( ) ( ) E S E E Va( ) ( ) ES ( ) 4
Q3. Please pot out a ch-squae adom vaable wth k degees of feedom coespods to whch patcula gamma dstbuto. Please wte dow the pdf, mgf, mea, ad vaace of a geeal gamma dstbuto ad of a ch-squae adom vaable wth k degees of feedom. Soluto Let W ~ k. W s deed a specal adom vaable. Gamma dstbuto ~ gamma(, ) (Some books use =α, ) x f ( ), 0 f x x e x ( ) o x ( ), 0 f x x e x () f s a o-egatve tege, the ( ) ( )! x f x dx x e dx 0 ( ) () () x x e dx 0 M t t ( t) M () t t 5
E( ) Va( ) Specal case : whe k, ~ k Specal case : whe ~ exp( ) Revew ~ exp( ) x p.d.f. f ( x) e, x 0 m.g.f. M () t t Q4. Let.. d. ~ exp( ),,,. What s the dstbuto of Soluto? M ( t) M ( t) t ~ gamma(, ) Q5. Let W ~ k. What s the mgf of W? Soluto MW () t t k k MW () t t 6
Q6. Let ~ W k, W ~ k, ad W ad W ae depedet. What s the dstbuto of W W? Soluto M ( t) M ( t) M ( t) W W W W k k t t t k k W W ~ k k. Moe geeal udestadg of samplg ad the sample mea dstbuto. Defto: Samplg eo s the eo esultg fom usg a sample to fe a populato chaactestc. Example: We wat to estmate the mea amout of Peps-Cola -oz. cas comg off a assembly le by choosg a adom sample of 6 cas, ad usg the sample mea as a estmate of the mea fo the populato of cas. Suppose that we choose 00 adom samples of sze 6 ad compute the sample mea fo each of these samples. These 00 values of wll dffe fom each othe somewhat due to samplg eo, but the values should all be close to -oz. Defto: Fo a adom vaable, ad a gve sample 7
sze, the dstbuto of the vaable,.e., of all possble values of, s called the samplg dstbuto of the mea. Ths pobablty dstbuto s a set of pas of umbes. I each pa, the fst umbe s a possble value of the sample mea, ad the secod umbe s the pobablty of obtag that value of the mea occu whe we select a adom sample fom the populato. Popetes of the Samplg Dstbuto of the Mea: ) Fo samples of sze, the expectato (mea) of, equals the expectato (mea) of.. I othe wods, ) The possble values of cluste close aoud the populato mea fo lage samples tha fo smalle samples. I othe wods, the lage the sample sze, the smalle the samplg eo. I patcula, the stadad devato of the samplg dstbuto of the meas,, wll be smalle tha the populato stadad devato,. I patcula, we have sze., whee s the sample Example: We ca easly lst the samplg dstbuto of the mea oly whe both the populato sze ad the sample sze ae vey small. Suppose that a pofesso gves a eght-pot quz to a class of fou studets. Let the class be the populato, of sze N = 4. Suppose the scoes ae, 4, 6, ad 8. We ca easly calculate the populato mea ad the populato stadad devato, usg the fomulae 4 6 8 gve chapte 3. We have 5, ad 4 5 4 5 6 5 8 5.36. 4 The populato dstbuto s dscete ufom;.e., f we adomly select oe membe fom the populato, we ae equally lkely to fd x =, x = 4, x = 6 o x = 8, whee the adom vaable s the scoe of ou adomly selected 8
studet. The pobablty dstbuto fo s theefoe x 4 6 8 P( = x) 0.5 0.5 0.5 0.5 Suppose, ow, that the pofesso does ot wat to calculate the populato mea usg the above fomula, but wats to estmate the populato mea usg a sample of studets. Thee ae seveal such samples whch could be selected. How may? If we sample wth eplacemet;.e., f we allow the possblty that the same studet may be selected moe tha oce fo a sample, the the umbe of possble samples s (ode of selecto s mpotat hee) 6. These ae {, }, {, 4}, {, 6}, {, 8}, {4, }, {4, 4}, {4, 6}, {4, 8}, {6, }, {6, 4}, {6, 6}, {6, 8}, {8, }, {8, 4}, {8, 6}, {8, 8}. If we the compute the sample mea fo each oe of these samples, we fd the samplg dstbuto of the mea lsted the followg table. Fequecy Relatve x Fequecy 0.065 3 0.5 4 3 0.875 5 4 0.5 6 3 0.875 7 0.5 8 0.065 Thus, f the pofesso adomly selects a sample of sze fom the class, she wll have a 0.065 pobablty of obtag a sample mea x =, a 0.5 pobablty of obtag a sample mea x = 3, ad so foth. The most lkely value of the sample mea s 5, wth a pobablty of 0.5. The expectato of may be easly calculated fom the above table. We get: 3 3 4 4 4 5 5 5 5 6 6 6 7 7 8 5. 6 Fo the stadad devato, we obta 9
5 3 5 3 5 7 5 7 5 8 5 6.58.36 We fd that.58 popety gve above., cosstet wth Defto: The stadad devato of the samplg dstbuto of the mea s called the stadad eo of the mea. Popety says that fo a gve populato, ad a gve adom vaable defed fo the membes of that populato, the stadad eo of the mea s smalle fo lage sample szes. Fo example, assume that ou populato stadad devato s. If ou sample sze s 4, the the stadad eo of the mea s 0.5. If ou sample sze s 6, the the stadad eo of the mea s 0.5. If ou sample sze s 00, the the stadad eo of the mea s 0.0 The followg theoetcal esult fom pobablty theoy s fudametal fo ou wok statstcal feece. The Cetal Lmt Theoem: Fo lage ( 30) sample szes, the adom vaable has a appoxmate omal dstbuto, wth mea ad stadad devato. I othe wods, the adom vaable Z has a appoxmate stadad omal dstbuto. Ths theoem holds egadless of the type of populato dstbuto. The populato dstbuto could be omal; t could be ufom (equally lkely outcomes); t could be stogly postvely skewed; t could be stogly egatvely skewed. Regadless of the shape of the populato dstbuto, the samplg dstbuto of the mea wll be 0
appoxmately omal fo lage sample szes. 3. The Samplg Dstbuto of the Sample Popoto Assume that we have a (lage) populato, whch s dvded to two subpopulatos. I oe subpopulato, each membe possesses a ceta chaactestc; the othe subpopulato, each membe does ot possess ths chaactestc. Assume that the popoto of membes of the ete populato who possess the chaactestc s p. We select a smple adom sample of sze fom the populato. We ae teested the popoto of the membes of the sample who possess the chaactestc of teest. Ths popoto s called the sample popoto, deoted by p. The Cetal Lmt Theoem tells us that, f the sample sze s lage, the a) The shape of the samplg dstbuto of p s appoxmately omal, havg mea p ad stadad devato p( p), povded p( p) 0. b) Ude the same codtos, the dstbuto of p p Z = p( p) s appoxmately stadad omal.